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On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics (1508.06149v1)
Published 25 Aug 2015 in math.AP
Abstract: We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for [ u_t = u \Delta u + u \int_\Omega |\nabla u|2 ] in bounded domains $\Omega\subset\mathbb{R}n$ and prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. We show that in this case the blow-up set coincides with $\overline{\Omega}$, i.e. the finite-time blow-up is global. Key words: Degenerate diffusion, non-local nonlinearity, blow-up, evolutionary games, infinite dimensional replicator dynamics